Throughout this section, let \((M,\omega )\)  be a compact connected symplectic surface with boundary \(\partial M.\)  By
\(\mathrm {SDiff}(M)\)  we denote the Lie groupSee [1]} for details on Lie groups and Lie algebras in an infinite-dimensional setting.
of all symplectomorphisms of \(M.\)  Note that all elements of \(\mathrm {SDiff}(M)\)  preserve the boundary \(\partial M\)  but do
not necessarily preserve the boundary \(\partial M\)  pointwise. The group \(\mathrm {SDiff}(M)\)  has the Lie algebra \(\mathfrak {s}\mathfrak {vect}(M)\)  of divergence-free vector fields on \(M\)  tangent to the boundary \(\partial M.\)  The regular dual space \(\mathfrak {s}\mathfrak {vect}^*(M)\)  can be identified with the space of cosets \(\Omega ^1(M)/d\Omega ^0(M)\)  (see Appendix). Moreover, the natural action of the group \(\mathrm {SDiff}(M)\)  on the space of cosets \(\Omega ^1(M)/\mathrm {d} \Omega ^0(M)\)  by means of pull-backs coincides with the coadjoint action of the group of symplectomorphisms \(\mathrm {SDiff}(M):\)
\(\mathrm {Ad}_\Phi ^*[\alpha ]=[\Phi ^*\alpha ],\) 
